Does it? The spell does not mention a Str check. If you create an illusion of a wooden wall over a doorway and an Ogre tries to ram that wooden wall in an attempt to break it, the Ogre would just go through the doorway because the wall isn't there. Granted, a given DM might still make this an Intelligence check action and if the Ogre makes it, the spell ends, and if the Ogre does not make it, he thinks that the wooden wall moved out of his way when he rushed it and then moved back, but it wouldn't require a Str check. Just like the Ogre would fall through the bridge that is not actually there no check required, the Ogre would move through the wall that is not actually there no check required.
Str checks are not part of the spell. Trying to break through the illusion works every time. The illusion is not actually there.
Yes. Phantasmal Force says it treats the effect as though it is real. One example given is it thinks it is the type of damage you make it seem like. So if you were to create a fire effect using phantasmal force on a red dragon, it would take no damage because it knows it is immune to fire.
If you create a restraining effect and it misses its intelligence save, it would think its wings were restrained. It would try to use its strength to break out, since that is how it normally reacts to a restraining effect.
Illusion spells by their nature cannot cover every type of effect. The spell interpretation is clear that it treats the phantasmal effect as though it were real justifying everything it does. It may have to make a strength check every round, but it would still get one. It would make it seem as though the chains were still on its wings. Nowhere does it say in the phantasmal effect you get to automatically restrain things, take away its abilities automatically, or anything of the kind. That would make phantasmal force more powerful than 9th level spells. It don't work like that. It creates an effect that causes the creature to interact with it in an appropriate fashion if it misses its intelligence save.